Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908784185573376 |
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| author | Propp, Adrienne M. Actor, Jonas A. Walker, Elise Owhadi, Houman Trask, Nathaniel Tartakovsky, Daniel M. |
| author_facet | Propp, Adrienne M. Actor, Jonas A. Walker, Elise Owhadi, Houman Trask, Nathaniel Tartakovsky, Daniel M. |
| contents | Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation constraint from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By optimizing over the reproducing kernel Hilbert space norm while applying a maximum likelihood estimation penalty on kernel complexity, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface fracture networks and arterial blood flow. Our results show that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs Propp, Adrienne M. Actor, Jonas A. Walker, Elise Owhadi, Houman Trask, Nathaniel Tartakovsky, Daniel M. Machine Learning Numerical Analysis Mathematical Physics Computational Physics 90C70, 60G15, 05C90 Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation constraint from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By optimizing over the reproducing kernel Hilbert space norm while applying a maximum likelihood estimation penalty on kernel complexity, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface fracture networks and arterial blood flow. Our results show that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical. |
| title | Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs |
| topic | Machine Learning Numerical Analysis Mathematical Physics Computational Physics 90C70, 60G15, 05C90 |
| url | https://arxiv.org/abs/2506.02337 |